One fallback option is the UK would have to accept keeping Northern Ireland in an effective customs union with the EU.

This is Mr Tusk’s second visit to Ireland in three months.

He has already pledged that the EU will stand with the Republic on the Irish border issue and Dublin’s efforts to ensure frictionless passage of people and goods.

Northern Ireland’s position post-Brexit is holding up agreement on Britain’s exit terms and a transition deal.

Mrs May’s supporters in Government, the Democratic Unionists, are adamantly opposed to any settlement distinguishing Northern Ireland’s EU trading relationship from the rest of the UK’s post-Brexit, what some have termed a border in the Irish Sea.

On Thursday Mr Varadkar will also hold a separate meeting with the secretary general of the Organisation for Economic Co-operation and Development (OECD), Angel Gurria.

Discussions will cover the OECD’s latest economic survey of Ireland and the potential for deeper Ireland-OECD co-operation.

Meanwhile, Department for Exiting the EU minister Robin Walker is due to meet Gibraltar chief minister Fabian Picardo on Thursday, and Cabinet Office minister David Lidington is hosting a meeting of the Joint Ministerial Committee with representatives of the devolved administrations.